Spectral Theory in Non-Separable Banach Spaces: Generalizations, New Properties and Structural Extension

This article addresses the extension of classical spectral theory to the setting of non-separable Banach spaces (NSBS), a field that remains underdeveloped despite its increasing relevance in functional analysis and mathematical physics. While spectral theory in separable Banach spaces has been thoroughly investigated—particularly through the work of Fredholm, Riesz, and others—the transition to nonseparable spaces presents both technical and conceptual challenges. These arise mainly from the absence of countable dense subsets and orthonormal or Schauder bases, which are instrumental in formulating and proving classical theorems. The main objective of this work is to provide a rigorous and systematic framework for spectral theory in NSBS, focusing on the classification of spectral components, the generalisation of key theorems, and the topological and algebraic behaviour of operators acting on such spaces. We begin by recalling the foundations of spectral theory in the classical (separable) case and proceed to identify the structural modifications required in the non-separable context. A central part of the analysis is devoted to the residual and continuous spectra, whose properties exhibit significant deviations from their counterparts in separable settings. We propose new definitions, prove generalised versions of the spectral theorem, and characterise the structure of the spectrum under various topological assumptions. Particular attention is given to the stability of the spectrum under perturbations—both compact and non-compact—and to the conditions under which the classical spectral decomposition remains valid. Several illustrative examples are provided to highlight the influence of non-separability on spectral behaviour. The results presented here contribute to a deeper understanding of spectral phenomena in infinitedimensional analysis, particularly in cases where separability cannot be assumed. They offer theoretical tools for future work in operator theory, interpolation, partial differential equations, inverse problems, and the theory of operator algebras. This article thereby lays the groundwork for further mathematical developments in non-separable environments and suggests new directions for interdisciplinary applications.